More about mismatch

Or caring about carry

An extra thought outside the regular Wednesday-Friday schedule.

Like the zombies in a horror movie¹, mismatch risk just keeps coming back. I’ve discussed this before, but failed to bury the topic for good.

Mismatch risk in the WTP

So grab your shovels and pickaxes², we’re digging deep this time!

The DNB site has changed since I last looked. It does not actually mention mismatch risk. It talks about the “unintended sharing of interest rate risk” (NL: onbedoelde deling van renterisico via het overrendement) and defines it as the difference between intended and realized excess return (NL: overrendement). Investors tend to think of the relative risk between the liabilities and the matching portfolio, but DNB have taken a different but similar perspective. I reproduce their formulas

\(\begin{split} R^{O, int}_i &= y_i(R^R - R^c)=:y_i R^{O,int} & \text{(intended)} \\ O^{r}_i &= \frac{y_i V_i}{\sum y_i V_i} O^{r} =:y_iV_i R^{O,r} \quad & \text{(realized)} \end{split}\)

We’re using the notation introduced earlier³, new is that Rᴿ is the total return on growth assets. They write R for a percentage return, and O for a €-amount: to make them comparable we define the second R. Notice that we can cancel Vᵢ in the second row, and can relate returns that are not cohort-specific!

\(\begin{split} R^{O, int} &= (R^R - R^c) & \text{(intended)} \\ R^{O,r} &= O^r/V_Y = \frac{1}{V_Y}\left(CFR - \sum_i R^H_iV_i \right) \quad & \text{(realized)} \end{split}\)

since the realized excess return O is the total Collective Fund Return minus the return on the guaranteed annuities. Subtracting these gives the realized mismatch:

\(\begin{split} V_Y(R^{O,r} - R^{O,int}) &= CFR - V R^c -\sum_j x_j V_j(R^M_j-R^c) - V_Y (R^R - R^c) \\ &= \left( (CFR - (V-V_Y)R^c)-\sum_j x_j V_j(R^M_j-R^c) \right) - V_Y R^R \end{split}\)

Now assuming Rᴿ and Rᶜ are not sensitive to rates⁴—only Rᵢᴹ∼Δᵢδr are—the first-order sensitivity is still what we wrote last time ∂CFR/∂r - V_XΔ. But now there is also a weird carry term to account for?!

Reserves

Most terms above scale with total cohort capital V, growth exposure VY or income VX. However, CFR= A∙ROA (Return on Assets) scales with assets A on the other side of the balance sheet! Total cohort liabilities V may differ in general from the total assets A. Reserves (A-V) represent own equity of the fund and implicitly provide leverage for CFR.

Therefore, accounting adjustments to the reserves should be added/deducted from CFR before distributions O are made. (The reserve for future costs apparently has a very long duration.) The question—for DNB and accountancy experts—is whether these reserve adjustments contribute to the mismatch? That is, is the scope of the unintended risk sharing restricted to the participants, or does it include risks run by the fund itself? I couldn’t find it on the website (DNB).

I think the mismatch should include risks to the fund’s own equity. In that case, the sensitivity of reserves to changes in rates should be accounted for in ∂CFR/∂r—or they should be described explicitly as additional cohorts so that Vʹ=A. Reseve “cohorts” need own policies for hedging x and risk y, and cash (1-x-y)!

• The reserve for future costs could have x>1 if its duration is greater than any cohort!

• The operational reserve may be proportional to total assets, so should grow ~ROA (that is, according to the fund’s average x=VX/V and y=VY/V).

• Similarly for the solidarity reserve, although arguably it might have y=1 building up.

• There are other reserves but I am not an accountant.

Portfolios, Cash and Carry

This is relevant in constructing the (benchmark for the) portfolio to match the liabilities V_XΔ. We know the target rates sensitivity to hedge, and know we should include reserve sensitivity, but what allocation to give it, to benchmark the manager?

Assume now we have allocated the reserves, so Vʹ=A, and write VʹY and Vʹ_XΔ for new sensitivities. We see inside the brackets of the previous equation, the matching portfolio should earn the carry term (V-VY)ʹ Rᶜ=(A-VʹY)Rᶜ for its funded part, and so should have a €-allocation of A-VʹY (theoretically) invested in cash⁵. Then the benchmark is to earn the cash return on that allocation, and match Vʹ_XΔ as closely as possible with off-balance derivatives. Presto!

1. Cover image is of wicked-but-relatable witch known as Madame Mikmak. Which is what mismatch starts to sound like when you’ve had enough of it.

2. Also said to be effective if we run into zombies.

3. Namely, for cash return Rᶜ and cohort capital accounts Vⱼ exposed to risk yⱼ and matching xⱼ

\(R_j^H=x_jR_j^M + (1-x_j)R^c = R^c + x_j(R_j^M-R^c) \text{ and } V_Y=\sum_j y_jV_j \)

4. If the investment portfolio contains credits and ∂Rᴿ/∂r is not zero, this could be said to be intentional sensitivity so not relevant for mismatch.

5. Note this sets the benchmark for the matching portfolio, not its investment universe. The question where to allocate to credits—in the return portfolio or matching—is not settled. Ideally, they would be invested in the matching portfolio but the intentional spread return swapped with the return portfolio so it is included in Rᴿ.